Optimal. Leaf size=113 \[ \frac{4 e^{2 a+x (2 b+d)+c} \, _2F_1\left (2,\frac{1}{2} \left (\frac{d}{b}+2\right );\frac{1}{2} \left (\frac{d}{b}+4\right );e^{2 (a+b x)}\right )}{2 b+d}-\frac{8 e^{2 a+x (2 b+d)+c} \, _2F_1\left (3,\frac{1}{2} \left (\frac{d}{b}+2\right );\frac{1}{2} \left (\frac{d}{b}+4\right );e^{2 (a+b x)}\right )}{2 b+d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.272688, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {5511, 2251} \[ \frac{4 e^{2 a+x (2 b+d)+c} \, _2F_1\left (2,\frac{1}{2} \left (\frac{d}{b}+2\right );\frac{1}{2} \left (\frac{d}{b}+4\right );e^{2 (a+b x)}\right )}{2 b+d}-\frac{8 e^{2 a+x (2 b+d)+c} \, _2F_1\left (3,\frac{1}{2} \left (\frac{d}{b}+2\right );\frac{1}{2} \left (\frac{d}{b}+4\right );e^{2 (a+b x)}\right )}{2 b+d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5511
Rule 2251
Rubi steps
\begin{align*} \int e^{c+d x} \coth (a+b x) \text{csch}^2(a+b x) \, dx &=\int \left (\frac{8 e^{2 a+c+(2 b+d) x}}{\left (-1+e^{2 (a+b x)}\right )^3}+\frac{4 e^{2 a+c+(2 b+d) x}}{\left (-1+e^{2 (a+b x)}\right )^2}\right ) \, dx\\ &=4 \int \frac{e^{2 a+c+(2 b+d) x}}{\left (-1+e^{2 (a+b x)}\right )^2} \, dx+8 \int \frac{e^{2 a+c+(2 b+d) x}}{\left (-1+e^{2 (a+b x)}\right )^3} \, dx\\ &=\frac{4 e^{2 a+c+(2 b+d) x} \, _2F_1\left (2,\frac{1}{2} \left (2+\frac{d}{b}\right );\frac{1}{2} \left (4+\frac{d}{b}\right );e^{2 (a+b x)}\right )}{2 b+d}-\frac{8 e^{2 a+c+(2 b+d) x} \, _2F_1\left (3,\frac{1}{2} \left (2+\frac{d}{b}\right );\frac{1}{2} \left (4+\frac{d}{b}\right );e^{2 (a+b x)}\right )}{2 b+d}\\ \end{align*}
Mathematica [A] time = 1.54049, size = 159, normalized size = 1.41 \[ -\frac{e^{c-\frac{a d}{b}} \left (d^2 e^{\left (\frac{d}{b}+2\right ) (a+b x)} \, _2F_1\left (1,\frac{d}{2 b}+1;\frac{d}{2 b}+2;e^{2 (a+b x)}\right )+d (2 b+d) e^{d \left (\frac{a}{b}+x\right )} \, _2F_1\left (1,\frac{d}{2 b};\frac{d}{2 b}+1;e^{2 (a+b x)}\right )+(2 b+d) e^{d \left (\frac{a}{b}+x\right )} \left (d \coth (a+b x)+b \text{csch}^2(a+b x)\right )\right )}{2 b^2 (2 b+d)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.1, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{dx+c}}\cosh \left ( bx+a \right ) \left ({\rm csch} \left (bx+a\right ) \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -48 \, b d^{2} \int \frac{e^{\left (d x + c\right )}}{48 \, b^{3} - 44 \, b^{2} d + 12 \, b d^{2} - d^{3} +{\left (48 \, b^{3} - 44 \, b^{2} d + 12 \, b d^{2} - d^{3}\right )} e^{\left (8 \, b x + 8 \, a\right )} - 4 \,{\left (48 \, b^{3} - 44 \, b^{2} d + 12 \, b d^{2} - d^{3}\right )} e^{\left (6 \, b x + 6 \, a\right )} + 6 \,{\left (48 \, b^{3} - 44 \, b^{2} d + 12 \, b d^{2} - d^{3}\right )} e^{\left (4 \, b x + 4 \, a\right )} - 4 \,{\left (48 \, b^{3} - 44 \, b^{2} d + 12 \, b d^{2} - d^{3}\right )} e^{\left (2 \, b x + 2 \, a\right )}}\,{d x} + \frac{4 \,{\left (12 \, b d e^{c} +{\left (24 \, b^{2} e^{c} - 10 \, b d e^{c} + d^{2} e^{c}\right )} e^{\left (4 \, b x + 4 \, a\right )} -{\left (24 \, b^{2} e^{c} + 2 \, b d e^{c} - d^{2} e^{c}\right )} e^{\left (2 \, b x + 2 \, a\right )}\right )} e^{\left (d x\right )}}{48 \, b^{3} - 44 \, b^{2} d + 12 \, b d^{2} - d^{3} -{\left (48 \, b^{3} - 44 \, b^{2} d + 12 \, b d^{2} - d^{3}\right )} e^{\left (6 \, b x + 6 \, a\right )} + 3 \,{\left (48 \, b^{3} - 44 \, b^{2} d + 12 \, b d^{2} - d^{3}\right )} e^{\left (4 \, b x + 4 \, a\right )} - 3 \,{\left (48 \, b^{3} - 44 \, b^{2} d + 12 \, b d^{2} - d^{3}\right )} e^{\left (2 \, b x + 2 \, a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\cosh \left (b x + a\right ) \operatorname{csch}\left (b x + a\right )^{3} e^{\left (d x + c\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh \left (b x + a\right ) \operatorname{csch}\left (b x + a\right )^{3} e^{\left (d x + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]